Few-Body Syst
DOI 10.1007/s00601-015-0990-z
C. A. Bertulani
Tunneling of Atoms, Nuclei and Molecules
Received: 30 January 2015 / Accepted: 24 April 2015
© Springer-Verlag Wien 2015
Abstract This is a brief review of few relevant topics on tunneling of composite particles and how the
coupling to intrinsic and external degrees of freedom affects tunneling probabilities. I discuss the phenomena
of resonant tunneling, different barriers seen by subsystems, damping of resonant tunneling by level bunching
and continuum effects due to particle dissociation.
1 Particles Moving in Mysterious Ways
1.1 History
The tunnel effect is one of the most subtle phenomenon explained by quantum mechanics, responsible e.g.,
for the existence of stars and ultimately for the existence of life [1]. The first application of particle incursion
into classically forbidden regions was done in nuclear physics. Nuclei, such as 210 Po, emit α-particles by
tunneling through the Coulomb barrier. This process lacked physical explanation until George Gamow used
the tunneling theory to calculate α-emission half-lives [2,3]. Tunneling is now a well known physical process
that has been incorporated in our everyday lives due to the increasing miniaturization in electronics such as
microchips. There have been important Nobel Prizes related to the straightforward use of this effect in industry,
such as the tunnel diode [4] or the scanning tunneling microscope [5].
1.2 Resonant Tunneling
Resonant tunneling is a particular kind of tunneling effect, frequently applied to miniaturization such as the
resonant diode tunneling device. In its simplest form, resonant tunneling occurs when a quantum level in one
side of a barrier has an energy match with a level on the other side of it. If this occurs in a dynamical situation,
tunneling is enhanced. In certain situations, the transmission probability is equal to one and the barrier is
completely transparent for particle transmission. In the resonant diode tunneling device, two semiconductor
layers sandwich another creating a double-humped barrier which enables the existence of quantum levels
within. On both sides of the barrier, electrons fill a conducting band on two outside semiconductors. A potential
difference between these outer layers allows one conducting band to rise in energy. Resonant tunneling leaking
occurs to the levels inside the double-hump barrier [4]. The net effect is the appearance of a current which
can be fine tuned. Resonant tunneling devices are compact and allow a fast response because the tunneling
between the thin double-humped barrier is a very fast process.
C. A. Bertulani
Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 5429, USA
Tel.: +1-903-886-5882
Fax: +1-903-886-5480
E-mail: carlos.bertulani@tamuc.edu
C. A. Bertulani
Fig. 1 Left: tunneling of a composite particle through a barrier is often equivalent to the tunneling of a single particle through a
double-humped barrier. Right: a Feshbach resonance appears when a preferred tunneling is induced by the coupling of a reaction
channel to one or more intrinsic degrees of freedom
1.3 Composite Particles
Resonant tunneling is not constrained to a particle tunneling though a double-humped barrier. An equivalent
process occurs when a composite particle tunnels through a single barrier, if each subsystem composing the
particle is allowed to tunnel independently or when the interaction reveals a priori unknown energy states. Then
the process is equivalent to the tunneling of one single particle though a double-humped barrier, and resonant
tunneling proceeds through the appearance of pseudo, quasi-bound, states (see Fig. 1, left). This effect occurs
frequently in atomic, molecular and nuclear systems. A good example is the fusion of loosely-bound nuclei
[6]. It also relates to Feshbach resonances when a system uncovers a state in a (closed) channel which is not
the same (open) channel where it sits in. This happens because the coupling with at least one internal degree
of freedom helps the reaction to proceed via resonant tunneling (see Fig. 1, right).
In the next Sect. 1 give several examples of how intrinsic degrees of freedom manifest in subtle ways,
leading to tunneling enhancement or suppression. As the subject is very vast and general, I will concentrate on
examples that I had the opportunity to work with during the last decades. They range from particle to molecular
physics. Some common characteristics are easy to understand, others not so much. An example of the last case
if given for diffusion and dissociation of molecules in optical lattices.
2 Tunneling of Composite Particles
Most theoretical problems involving tunneling are not amenable to analytical solutions. There are several
computational methods available for them. Below I will just mention a very well-known method, used to
generate some of the calculations displayed in the figures of this review.
2.1 Stalking Microscopic Particles
It is interesting to study the time-dependence of tunneling, although it is prohibitive for most cases of interest
due to the long tunneling times, e.g., in nuclear α-decay processes. The time evolution of a wave function on a
space lattice can obtained by solving the Schrödinger equation by a finite difference method. The wave function
Ψ (t + Δt) can be calculated from the wave function at time t, Ψ (t), by applying the unitary time evolution
operator, U , i.e., Ψ (t + Δt) = U (Δt, t)Ψ (t) = exp [−i H Δt/h̄]Ψ (t), where H is the system Hamiltonian.
For a small time step Δt, one can use the unitary operator approximation, valid to order (Δt)2 ,
U (t + Δt) =
1 + (Δt/2i h̄)H (t)
.
1 − (Δt/2i h̄)H (t)
(1)
Other approximations for exp [−i H Δt/h̄] such as the Numerov algorithm, etc., can also be used. Any of these
numerical procedures requires carrying out matrix multiplications and inversions at each iteration and the
Tunneling of Atoms, Nuclei and Molecules
Fig. 2 Left: tunneling of a composite particle through a single barrier. Right a and b the effective potential for the center of mass
motion of the system
Fig. 3 Left: transmission probabilities for a two-body system through a rectangular barrier [12–14] as a function of the ratio of
the center of mass energy and the barrier height V0 . The solid line is the transmission probability in which the system remains
in the ground state and the dashed line is the for the case the system transits to an excited state. Right: transmission probability
of a molecule through a thin barrier as a function of the barrier width a times the molecule momentum. The dashed curve is the
transmission probability when the molecule is in an excited state after the transmission [17,18]
number of operations grows very fast with the number of points in the coordinate lattice [7,8]. It also increases
with the number of particles in the system at hand, becoming numerically demanding for large fermionic or
bosonic systems [9]. But the most difficult problem is related to the existence of times scales which differ
by a large amount. For example, during alpha decay the decay time is often much larger (e.g., 1030 ) than
the intrinsic time dependence of the alpha particle state in the nucleus. This makes it impossible to treat the
problem in a time-dependent fashion. But there are ongoing theoretical efforts to handle the time-dependent
problem with widely different time scales [10,11,15].
As an application of the method described above, I will present below results for the tunneling of a two-body
system through a simple rectangular barrier.
2.2 Transmission of Composite Particles
Assume a system of two bound particles with an attractive infinite square-well interaction of range d, being
transmitted through a rectangular barrier of width a and height V0 . By choosing the center of mass x =
(x1 + x2 )/2 and relative y = x1 − x2 + d/2 coordinates, it is straightforward to show that the center of mass
wave equation has an effective double-humped potential W (x, y). Figure 2 shows that for various values of
the relative distance y, the effective potential W is in fact a double-humped barrier [4,12–14]. This barrier can
hold energy states that yields resonant tunneling if they are equal to the center of mass energy, as shown in
Fig. 3 for the dependence of the ratio of the bombarding energy and the barrier height [4,12–14].
The simple example mentioned above can be readily applied to a very practical case; that of a bound
molecule transmitting through a barrier. This is illustrated below.
2.3 Transmission of Molecules
Another way to enhance tunneling is allowing the system to make a transition from an excited state to a
lower energy state. This has been shown analytically in Ref. [16] for a two-spin system tunneling though a
barrier. The transmission probability as a function of the barrier width also shows signs of resonant behavior, as
observed in calculations for transmission of molecules though barriers [17,18]. Figure 3 shows the transmission
C. A. Bertulani
Fig. 4 Left: a schematic rectangular barrier simulates the proton-nucleus interaction for a deuteron-nucleus fusion reaction. The
solid curve is the effective potential for the center of mass motion. Right: transmission probability for the deuteron, with (solid)
and without (dashed) consideration of coupling to intrinsic modes [16]
probabilities for a two-body system through a rectangular barrier [12–14] as a function of the ratio of the center
of mass energy and the barrier height V0 . The solid line is the transmission probability when the system remains
in the ground state and the dashed line is when the system transits to an excited state. The right hand side of
Fig. 3 shows the transmission probability of a molecule through a thin barrier as a function of the barrier width
a times the molecule momentum. The dashed curve is the transmission probability when the molecule is in an
excited state after the transmission [17,18]. By moving to its ground state the excited molecule increases its
chance of transmission. On the opposite trend, the opening of the higher channels decreases the probability of
transmission.
The two-body tunneling problem also encounters abundant applications in nuclear physics, involving
cluster-like objects. The simplest one is the deuteron, composed of a proton and a neutron forming a relatively
loose-bound system (E B = −2.224 MeV). Other common systems include those containing alpha particles.
Tunneling involves sudden accelerations leading to gamma-emission in alpha tunneling, as described below.
2.4 Fusion of Loosely Bound Nuclei
Loosely-bound systems also display interesting tunneling properties because they can end up in separate parts
by reflection or transmission through a barrier. If the separate parts see a different barrier then the effective
barrier for the center of mass tunneling can also be very different than the original one. A good example is the
fusion of the deuteron with another nucleus in which case the proton-nucleus potential has a Coulomb repulsion
part whereas the neutron-nucleus potential does not. As shown in Ref. [16] the net effect is an enhancement
of the transmission probability through the barrier (see Fig. 4). Fusion of exotic, neutron rich, nuclei due to
particle transfer, or tunneling, is now a very active area of research [19–26] (for a review, see Ref. [27]).
An intriguing question is how much time a particle takes to tunnel a barrier [28]. A Bremsstrahlung
measurement following α-decay [29] apparently paved the way to determine tunneling times by inspecting
the interference pattern in the radiation spectra. A time-dependent description of the emission process with
a pre-formed α-particle has shown that a model with a free, uncorrelated, α-particle within the nucleus is
incompatible with the observations for this process [30]. The spectrum of Ref. [29] was also explained in a
more traditional theoretical method [31–33]. However, the question related to tunneling times is still fascinating,
in particular when it comes to the possibility of time travel and the existence of time warps and worm holes
[34].
The emission of radiation during tunneling is also manifest in virtual processes, a fascinating discussion
of which is presented below.
2.5 The Quantum Münchhausen Effect
Additional interactions during tunneling can lead to other subtle effects, e.g., as proposed in Ref. [35]. Since
barrier transmission is proportional to an exponentially decreasing dependence with the particle energy, any
Tunneling of Atoms, Nuclei and Molecules
Fig. 5 Left: the quantum Münchhausen effect [35]. Right: enhancement of fusion for the reaction d + 3 He due to couplings to
real and virtual photons
small effect can change it dramatically. The so-called quantum Münchhausen1 effect assumes that a virtual
photon emitted in one side of the barrier and caught on the other side can enhance barrier penetrabilities (Fig. 5).
This is allowable in quantum mechanics because a particle has an extended wave nature. The particle kinetic
energy can momentarily increase (within bounds set by Heisenberg’s uncertainty principle) and tunneling can
be enhanced. A non-relativistic reduction of the particle-radiation interaction yields [35]
H Ψ (r) + Π (r, r ; E)Ψ (r )dr = EΨ (r), where Π (r, r ; E) = M(r, r ; E) + iΓ (r, r ; E). (2)
In this expression, M includes the self-energy, mass renormalization and virtual photons, while Γ includes the
effects of radiative decay width and real photons. This effect has been studied further in Ref. [37] and it was
shown to be very small indeed for tunneling of stable nuclei. But an account of this effect for loosely-bound
nuclei (and molecules) has not been explored at depth.
In atomic systems, electron tunneling is sometimes responsible for chemical bonding. But subtle effects
are also present in dynamical situations, in which electron tunneling can be the cause of increased stopping of
protons on a hydrogen target at extremely low energies, as discussed next.
2.6 Electron Hoping Induces Stopping
As discussed above, tunneling can be strongly affected by matching energy levels, the appearance of pseudolevels during the tunneling process (Feshbach resonances), and by energy transfer to the relative motion by
coupling to the internal degrees of freedom. Other effects appear when many energy levels are involved. Perhaps
the best and clearer examples appear in atomic physics. Stopping power, or the energy loss of a projectile per
unit length, S = −d E/d x, is an important quantity for experiments at low bombarding energies. The energy
at which a nuclear reaction occurs depends on the average energy loss of the projectile in an atomic target.
Correcting for the stopping power is thus imperative for low energy experiments. At extremely low projectile
energies, only a few phenomena are energetically viable such as the excitation of vibrations (plasmons) in
the medium for solid materials, Rutherford scattering and straggling, or charge exchange when one or more
electrons are transferred to the projectile.
Stoping by charge exchange in p + H and p + D reactions has been studied in Ref. [38]. Because of the
low bombarding energy, the electrons adjust themselves quickly and one can use the adiabatic approximation.
The potential felt by the electron resembles a symmetric Coulomb barrier around the two atomic centers.
As a function of the relative nuclear distance, the electronic energy levels smoothly undergo from hydrogen
to helium levels (left side of Fig. 6), passing by molecular orbitals at intermediate distances. Expanding the
electronic wavefunction as a time admixture of such orbitals, a set of coupled-channel equations is obtained
with help of the Hellmann–Feynman relation. As the bombarding energy decreases only the two lower levels,
1sσ and 2 pσ play a relevant role. The electron jumps back and forth between the two atoms by tunneling
through the symmetric Coulomb barrier. The right hand side of Fig. 6 shows the electron exchange (tunneling)
1 Baron Münchhausen is supposedly the greatest liar ever depicted in fiction literature and in movies. He once explained how
he could save himself from sinking in quicksand by pulling his own hair up [36].
C. A. Bertulani
Fig. 6 Left: energy levels of an electron as a function of the distance between a proton and a hydrogen atom. Right: the exchange
probability as a function of the impact parameter for two proton kinetic energies
probability as a function of the collision impact parameter in atomic units and for two bombarding energies: (a)
10 KeV, and (b) 100 eV. One notices that at higher energies the tunneling probability increases as the distance
decreases mainly due to the decrease of the barrier height and the resonant tunneling effect. When the energy
is very low, as seen in the lower panel, the electron jumps frenetically back and forth between the two atoms
due to resonant tunneling. At very low energies the exchange probability as a function of impact parameter is
given in terms of the simple formula [38]
∞
1
1 1
Pexch = + cos
(3)
E 2 p (t) − E 1s (t) dt .
h̄ −∞
2 2
The dashed curve in Fig. 6 proves that the prediction based on this equation holds very well, specially at very
low energies. It is expected to be exact
∞ at extremely lowenergies. The probability minima occur for impact
parameters satisfying the relation −∞ E 2 p (t) − E 1s (t) dt = 2π h̄(n + 1/2), a relation more than familiar
to all of us. It is the interference between the 1sσ and the 2 pσ states that induces oscillations in the exchange
probability.
The case discussed above involves tunneling of only one electron. A more interesting and rather complex
situation arises when two electrons are present, as in the case of stopping of protons on a helium gas, as
discussed below.
2.7 Damping of Resonant Tunneling
The case of p + He is more interesting, because of the two active electrons. One can expand the electron
wave functions in terms of the two-center basis and obtain a set of time-dependent Hartree–Fock equations
[39]. In contrast to the p + H or p + D case the much larger number of levels make a difference. Due to level
repulsion, the levels tend to bunch as seen in the left panel of Fig. 7. The transition between the lowest levels
shows an oscillating behavior due to resonant tunneling between one electron in the ground state of hydrogen
and the first excited state of helium. But the oscillations are damped. This is due to the interference between
the low-lying states and a bunch of states of average energy E and width 2Γ , as shown on the left side of
Fig. 7. The damping can understood with the Landau–Zener theory for level crossing. At the crossing there is
a probability of an adiabatic transition where (1 − Pexch ) given by the Landau–Zener formula
2
2π Hss
Pexch = exp
,
(4)
h̄vd(E s − E s )/d R
Tunneling of Atoms, Nuclei and Molecules
Fig. 7 Left: adiabatic energies (1 a.u. of energy = 27.2 eV, 1 a.u. of length = 0.53 Å) for the electronic orbitals of the (H − He)+
system as a function of the internuclear separation. As the atoms approach each other slowly, curves of same symmetry repel
each other. A transition between states s and s’ can occur in a slow collision. In a fast collision a diabatic transition, with the
states crossing each other, will occur. This is shown in the inset. Right: probability of charge exchange in the collision p + 4 He
showing the damped resonant behavior as a function of the impact parameter and for proton energy E p = 10 keV
where v is the collision velocity, Hss is the off-diagonal matrix element connecting states s and s and the term
in the denominator is the derivative of the energy difference with respect to the separation distance between
the two nuclei. The interference with the neighboring states introduces a damping in the charge exchange
probability,
2πΓ 2 b
2 E b
Pexch = cos
,
(5)
exp −
h̄v E
2h̄v
where E ≈ 1 a.u. is the average separation energy between the 0 level and the bunch of higher-energy
levels shown in Fig. 7. The exponential damping factor agrees with the numerical calculations if one uses
Γ = 5 eV. The exchange probability for the electron drops sharply to zero when the energy of the proton
is smaller than 18.7 eV. This is easy to understand because this is the energy necessary for the transition
1s 2 (1 S0 ) → 1s2s(3 S) in He. Only when this transition occurs, resonant tunneling will be possible between
the ground state in hydrogen and the excited state in helium.
Finally, we consider what happens in tunneling of composite objects through multiple barriers. As an
illustrative example, we describe below the case of diffusion and dissociation of molecules in optical lattices.
Despite the theoretical simplicity of the system under consideration, the complexity is much larger than in the
previous examples and some open questions remain.
2.8 Diffusion in Optical Lattices
Using standing wave patterns of reflected laser beams optical lattices can be built to study the diffusion
of ultracold atoms and molecules [40–43]. Fundamental aspects of diffusion of composite objects, such as
molecules, subject to transforming transitions from bound to continuum states can be studied in these optical
lattices [44]. In Ref. [44] the confining potential described by the optical lattice potential has been assumed to
be a sine function with a periodicity D, equal to half the wavelength of the interfering laser. The Hamiltonian
for the molecule is given by H = T1 + T2 + V1 + V2 + v, where Ti is the kinetic energy of atom i, Vi is the
periodic potential of the lattice and v is the interaction potential between the atoms for a diatomic molecule. A
loosely bound, Rydberg-like, system of molecules diffusing through the lattice molecule was studied in Ref.
[44].
C. A. Bertulani
Fig. 8 Left: lattice diffusion of molecules. The hatched histograms are the relative probabilities of finding a molecule in its ground
state at a given position along the lattice. The solid histograms give the relative probabilities of finding individual atoms after
the dissociation. The molecules are initially confined in the site at the origin. Right: time dependence of the spreading width of
bound molecules, σ M (t), shown by the solid line, and of dissociated atoms, σ A (t), shown by the dotted line. The dashed curve
is a fit of the asymptotic time dependence with the analytical formula, Eq. (6)
The molecules tend to dissociate as they tunnel through the barrier when their binding energy decreases. At
very small binding energies, the trend is reversed and the molecule tends to remain intact. As shown in Fig. 8,
loosely-bound molecules are resilient to breakup: even at the smallest binding value, the breakup probability
is not one. This result is not unusual, as other tunneling systems exhibit similar behavior. This has been proven
for the familiar example of tunneling of Cooper pairs [45]. The pair does not usually dissociate as it tunnels
through a barrier. However composite particles may dissociate as the particles tunnel through multiple barriers,
as in the case of a lattice potential. It takes time for the molecules to dissociate and additional time for the
atoms to diffuse after the dissociation. Therefore, the molecules initially confined within one pocket of the
lattice will tunnel and diffuse away from the initial position at a speed higher than that of the atoms that are
created in the dissociation of the molecules.
In Ref. [44] a diffusion equation
in an optical lattice in terms of the diffusion velocity of the
for molecules
2 2
wave packet was given by σ (t) = r (t) , where r (t) is the expectation value of the square of the position
of the molecule calculated from the solution of the Schrödinger equation. The numerical results show that after
a transient time, and for a strongly bound molecule of mass m, the approximate result holds,
1/2
h̄ 2 t
σ (t) = C
,
(6)
mbσ02
where C and b are constants depending on the parameters defining the lattice and σ0 = σ (t = 0). b has the
meaning of a “viscosity” generated by the wiggling potential dependence of the lattice. The linear behavior in
Eq. (6) is shown by the dashed line in Fig. 8 (right). Also shown are the numerical results for the same quantity
(solid line), and the diffusion velocity of individual atoms in the lattice (dotted line).
A diffusion coefficient can be defined by D = (∂σ 2 /dt)/2. Using Eq. (6) one gets
C
h̄ 2
D=
.
(7)
2 mbσ02
This result deviates from the classical Einstein diffusion constant, D = kT /b. The effective “temperature”
for this system is the kinetic energy of the initial state, T h̄ 2 /mσ 2 (t = 0). The dependence of the temperature,
or diffusion coefficient, on the initial kinetic energy is a result which needs to be verified experimentally. The
molecules are assumed not to interact. Therefore these results might not hold when molecules interact in a
condensate. The time dependence of the spreading width of the dissociated atoms within the lattice is not
well fitted by any power-law dependence on time, as shown in Fig. 8. The reason for this behavior is not yet
understood.
Tunneling of Atoms, Nuclei and Molecules
3 Conclusions
In this short review I have presented an overview of tunneling of composite objects. I started by giving an
example of how one would tackle the problem computationally, followed by an application of the method to
the tunneling through a rectangular barrier. It is evident that coupling to open and closed channels lead to a
clear resonant behavior visible in the tunneling probabilities.
I went on to discuss a few applications of tunneling of composite objects to real situations, such as
molecules passing through thin membranes, cluster-like nuclei tunneling during a nuclear fusion process, and
the emission of radiation during tunneling as, e.g., observed during alpha-decay. Other examples were given
in atomic physics where tunneling can enhance or suppress the stopping of particles at very low energies. This
important process leads to corrections for the energy loss of ions in reactions of astrophysical interest. Finally,
the more difficult theoretical treatment of tunneling of composite objects through multiple barriers has been
illustrated with the example of molecules diffusing in optical lattices.
Although a (large) number of theoretical works have studied tunneling phenomena in various situations,
quantum tunneling of a composite particle in which the particle itself has an internal structure, has yet to be
fully clarified.
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